Prof. Challib on the Principles of Hydrodynamics. 29 



law of pressiu-e does not rest on the same evidence for fluid in 

 variable motion as for fluid at rest. It vsill appear, however, by 

 the following demonstration, that the evidence in the fonner 

 case may be made identical with the evidence in the other by the 

 introduction of a principle which may be regarded as an axiom, 

 viz. that if a common velocity, or common increments of velocity, 

 be impressed on all the parts of the fluid and on the containing 

 sohds in the same direction, the density and pressiu'e of the 

 fluid remain unaltered. 



Proposition II. The pressure at a given instant at any point 

 in the interior of fluid in motion on an imaginary plane passing 

 through that point, is the same whatever be the position of the 

 plane. 



Conceive the velocity of the fluid particle which is at a given 

 point at a given time to be impressed upon it and upon all the 

 parts of the fluid and its containing solids in a direction opposite 

 to that in which it takes place. The particle is thus reduced to 

 rest. If also its effective accelerative force at each succeeding 

 instant be impressed on all the parts of the fluid and the con- 

 taiuing sohds in the direction contraiy to that in which it takes 

 place, the particle will remain at rest. By the principle stated 

 above, the relative positions of the particles of the fluid and the 

 pressui-es at all points are in no respect changed by thus im- 

 pressing a common velocity and common accelerative forces in 

 common directions. Since, then, the particle under considera- 

 tion continues at rest, we may apply to it the same reasoning as 

 that applied to a given elementary portion of the fluid in Propo- 

 sition I. ; the only difference being, that as in that proposition 

 the pressure is supposed to be vmiform through an indefinitely 

 small space, in the proposition before us it is supposed to be 

 miiform both through an indefinitely small space and dui-ing an 

 indefinitely short time, because the density of the given particle 

 is continually changing. The eiTors of these suppositions are 

 infinitesimal quantities of an order that may be neglected. Hence 

 the law of pressure results precisely as in Proposition I., the 

 efi"ective accelerative force being supposed to be always finite. 

 Being shown to be true of any single particle at any time, it is 

 true of all particles at all times. Consequently the law of equal 

 pressure in all directions from a given point has been proved to 

 hold generally both in fluid at rest and in fluid in motion. 



It may be here remarked, that the i)rinciple of impressing a 

 common velocity and common accelerative forces in a common 

 direction, on all' the material points concerned in any instance 

 of motion, gives the means of converting a dynamical into a sta- 

 tical problem, and is in fact the most elemental^ foi-m of D'Alem- 

 bert's principle. 



